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Asymptotic Problems in Number Theory Summary of Lectures- Spring 2015

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Asymptotic Problems in Number Theory Summary of Lectures- Spring 2015 Math 719: Asymptotic Problems in Number Theory Summary of Lectures- Spring 2015. In this document I will give a summary of what we have covered so far in the course, provide references, and given some idea of where we are headed next. 1 Lecture 1 In the first lecture I gave an overview of what we will cover in the first half of the course. The broad goal is to understand the behavior of families of arithmetic objects. We want to understand what an `average' object looks like, and also what an `extremal' object looks like. We will look at this question in a few different settings. The three main topics will (likely) be: 1. Class groups of imaginary quadratic fields; 2. Curves over finite fields; 3. Linear codes. Our first major goal is to understand positivep definite quadratic forms of given discriminant and then connection to class groups of the field Q( −d), or really more accurately, the quadratic ring of discriminant −d. We will spend approximately the first three weeks setting up this connection, first focusing on quadratic forms and then transferring over to ideal class groups. p As we vary over all quadratic imaginary fields K = Q( −d) where d > 0 is squarefree, we get some set of finite abelian groups C(OK ). What can we say about these groups? How large are they on average? How often is this group trivial? What can we say about the p-parts of these groups for fixed p? Understanding these questions will require using some of the key ideas of class field theory and also understanding some analytic techniques. To answer the question about average size we will introduce the Dedekind zeta function of a number field and discuss Dedekind's class number for- mula. The Stark-Heegnerp theorem gives a complete answer to the second question, a complete list of d such that Q( −d) is a unique factorization domain. For the third question we will investigate the Cohen-Lenstra heuristics, which roughly state that a group should arise in proposition to the inverse of its number of automorphisms. The prime p = 2 behaves differently from odd primes with respect to p-parts of class groups. We will discuss genus theory, both on the quadratic forms side and on the ideal class group side in order to understand this special behavior. The only other proven case of the Cohen-Lenstra heuristics is the average number of 3-torsion elements in C(OK ). This comes from counting cubic fields of 1 bounded discriminant and a little bit of class field theory. We will prove this result, the Davenport- Heilbronn theorem, and discuss related questions about counting number fields and the orders they contain. 2 Lecture 2 In the first half of this lecture we discussed the other two main topics that we plan to cover in this course. How many points can a smooth genus g curve over a finite field Fq have? We call this number Nq(g). We can try to compute this number for particular values of (q; g), or we can try to understand its asymptotic behavior. We consider the quantity N (g) A(q) = lim sup q : g!1 g We will discuss elliptic curves over finite fields, the case g = 1, where we know Nq(1) for all q. We can also try to understand the `average' behavior of all elliptic curves over a fixed field Fq. As q ! 1 work of Birch, building on fundamental results of Deuring in the theory of complex multiplication gives a nice answer. This is the `vertical' Sato-Tate theorem for elliptic curves. We will give a detailed sketch of the proof. p Hasse's theorem tells us that an elliptic curves over Fq has q +1−t rational points, where jtj ≤ 2 q. We can see that this cannot be improved. The Hasse-Weil bound is the analogue for higher genus curves, and here there are interesting and subtle questions about upper bounds for Nq(g). We will discuss zeta functions of curves, the Weil conjectures for curves, and improvements to the Hasse-Weil bound. We will also discuss the cases of g = 2 and g = 3. In a different direction, we will discuss curves like the Hermitian curve over Fq2 which has very many rational points and a very large automorphism group given its genus. We will also discuss the theorem of Ihara / Tsfasman-Vladut-Zink which gives an asymptotic lower bound for A(q) by studying rational points on modular curves over finite fields. We also introduced some basic problems in coding theory. How large can a code over Fq of length n be if the minimum distance of the code is at least d? What if we restrict to linear codes? We will discuss both upper bounds and lower bounds. These lower bounds will often come from algebraic constructions, evaluating vector spaces of polynomials at specified points. In particular, we will discuss algebraic geometry codes, and more specifically the Goppa codes that generalize the classical Reed-Solomon and Reed-Muller codes. These codes come from taking an algebraic curve and a divisor D, and evaluating every polynomial in the Riemann-Roch space L(D) at a specified set of points. For the rest of this lecture we shifted to talk about the basics of integral binary quadratic forms. For this part of the course (approximately the first three or four weeks) I will closely follow parts of Cox's book Primes of the form x2 + ny2. It is not available as an online resource for Yale students, but I hope that you are able to get a copy. 2 In this lecture we first introduce some of the basic terminology of binary quadratic forms. I closely followed 2.A of Cox for this. We basically did everything up to Theorem 2.8, but also phrased quadratic forms in terms of matrices. 3 Lecture 3 In lecture 3 we started by introducing reduced quadratic forms. I sketched the proof of Theorem 2.8 of Cox, that every primitive positive definite form of discriminant D < 0 is properly equivalent to a reduced one. An easy consequence of the definitions is that there are only finitely many reduced forms of given discriminant. This number is called the class number, or `form class number' h(D). We call the set of all classes C(D). Later we will show that this is a group. We then defined the genus of a quadratic form of discriminant D in terms on the set of values ∗ represented by (Z=DZ) . We introduced the homomorphism χ, which is defined in Lemma 1.14 of Cox. The results about genera we talked about are in Cox 2.C. In particular, we proved Theorem 2.16. We then sketched Landau's argument for the set of all D ≡ 0 (mod 4); D < 0 such that h(D) = 1. This is Theorem 2.18. We gave two examples of genera and showed how when each genus consists of a single class we get nice corollaries about representation of integers by quadratic forms. The examples we gave are on page 30 of Cox. We defined the principal form and sketched ∗ the proof of Lemma 2.24, showing that the values in (Z=DZ) represented by the principal genus form a subgroup, and that every other genus gives a coset. We stated the fact that the map taking a class to the coset represented by its genus is a group homomorphism, but in order to make that rigorous we need C(D) to be a group. Once you believe that C(D) is a group, Lemma 3.13 shows that Dirichlet composition makes this map into a group homomorphism. 4 Lecture 4 We started with a review of everything about forms of discriminant D up to this point. We then defined Gauss composition and stated that it makes C(D) into a group. We defined Dirichlet composition, which is much more explicit and easier to compute with. This is all done in the first section of 3.A in Cox. There are a lot of details to check, but it is worth going through and checking some of this at least once. In the last part of the lecture we discussed Bhargava's work on Gauss composition using 2 × 2 × 2 cubes. I first talked about how to slice a cube in three different ways, which gives three pairs of 2 × 2 matrices. For each pair we get a quadratic form. It is a fact that I did not prove that each of these forms has the same discriminant. You can just check this by writing down the forms explicitly in terms of the eight entries of the cube and actually taking the determinants. 2 n 2 I described another way of thinking about this in terms of an explicit basis of Z ⊗ Z ⊗ Z so that a cube gives an element of this space. We have an action of G = SL2(Z) × SL2(Z) × SL2(Z) 3 on this space, and so we also have an action on cubes. We need only understand the action by γ × I × I (and the permutations of this) where we extend things in the obvious way. So an element of G acts on a cube, and an element of G also acts on our triple of quadratic forms. We stated a major theorem, that this action `commutes'. We also stated the correspondence between G-orbits on this vector space of cubes and the set of isomorphism classes of pairs (S; I1;I2;I3) where S is a quadratic ring and (I1;I2;I3) are a balanced triple of `oriented ideals' of S.
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